Magnus C. Ørke: Highest waves for fractional Korteweg–De Vries and Degasperis–Procesi equations

Abstract:

In recent years global bifurcation theory in combination with precise regularity estimates has been used to prove existence of specific classes of traveling wave solutions for nonlinear and nonlocal PDEs. I will discuss this method for a class of fractional Korteweg–De Vries and fractional Degasperis–Procesi equations with an inhomogeneous Fourier multiplier of parametrized order in the range of \((-1, 0)\), and show that there are highest, cusped traveling-wave solutions for both equations, characterized by optimal Hölder regularity exactly attained in the cusp.